
associated to the map \({M}_{g,2} \rightarrow {M}_{g,1}\)  with fiber
\(C^{\circ }\) .
We may obtain this from the Birman exact sequence [1]} for mapping class groups,
after identifying the fundamental group for \(M_{g,n}\)  with the mapping
class group of an \(n\) -times punctured genus \(g\)  surface.
(The case \(n = 0\)  follows from contractibility of the universal cover of
\(M_g\)  [1]} with covering group given by
the mapping class group,
and the case of general \(n\)  can be deduced from the Birman exact sequence
[1]}.)
